In the video, Weber presses a hand-written note against the cabin window.
Antoine, confirming Weber's death in a Facebook post, said he "never emerged from those depths".
The Manta Resort said the victim "tragically drowned while free diving alone outside the underwater room."
Weber and Antoine had booked four nights at the resort's underwater room, which lies approximately 250m (820ft) from the shore.
The water where the cabin is anchored is around 10m (32ft) deep.
On the third day of their stay, Weber, from Baton Rouge, Louisiana, plunged into the water to hold his proposal note against the cabin's glass window.
His note read: "I can't hold my breath long enough to tell you everything I love about you. But ... everything I love about you, I love more every day!"
Later in the video, Weber turns over the sheet of paper to reveal the proposal, before pulling an engagement ring from his shorts and swimming out of view.